The present disclosure relates generally to multivariable input/output control designs and, more particularly, to a multivariable controller design method for multiple input/output systems with multiple input and/or output constraints.
All practical control systems have constraints on the inputs or control actuators of the system (e.g., due to hardware limitations). Moreover, there are often multiple state/output constraints on the system as a result of certain safety and/or operability concerns. The majority of industrial control system applications with input/output constraints are generally limited to single-input single-output (SISO) systems or multi-input multi-output (MIMO) systems with low input-output interaction. Most of such controllers include the use of integral action to account for constant modeling errors and unknown disturbances. However, it is well known that the presence of such integral actions lead to the “windup” phenomenon in the presence of actuator constraints, wherein the control actuator saturates at the max/min limit, while the integral action remains active and builds up, leading to performance degradation, or worse, instability. A common approach to overcome this issue due to actuator (input) constraints is to simply terminate the integral action (i.e., stop updating the integral term) upon actuator saturation.
Although this simple approach is often sufficient for SISO systems with actuator saturation, its straightforward extension to MIMO systems with multiple actuators and corresponding minimum/maximum (min/max) constraints by individually limiting the corresponding integral term on each actuator has serious limitations. In particular, multivariable controllers for MIMO systems rely on the coordination between the multiple actuators to achieve the desired control objectives. However, the individual saturation of the integral term on each actuator does not account for the desired coordinated interaction desired between all the actuators. Thus, this simple technique of individually limiting the integral action on each actuator is often inadequate for MIMO systems, and leads to significant performance loss or even instability. More sophisticated techniques for providing “anti-windup” or “windup compensation” for MIMO systems with multiple actuators and min/max saturation limits employ the use of an “anti-windup” controller that is active only when the actuators saturate at a min/max limit. More specifically, for a general MIMO nonlinear system with a state-space description of the form:{dot over (x)}=ƒ(x,u)y=h(x,u)  Eq. 1with states xεn, control inputs (actuators) uεmu, and controlled outputs yεmy (considering only square control systems with mu=my), a dynamic multivariable controller with integral action and anti-windup protection has a following general form:{dot over (x)}c=ƒc(xc,x,y)+L(u−sat(u))u=hc(xc,x,y)+M(u−sat(u))  Eq. 2with the controller states xcεnc and anti-windup gains L & M. These anti-windup gains act upon the difference (u−sat(u)), wheresat(u)=u, if umin<u<umax=umin, if u≦umin=umax, if u≧umax  Eq. 3denotes the standard saturation operator. As can be seen, the anti-windup terms are active only on saturation when one or more of the actuators u saturate at a min or max limit. In most cases, the anti-windup gain M is set to 0, while the anti-windup gain L is designed to enforce performance/stability in the presence of actuator saturation. The design of such multivariable controllers with multivariable anti-windup protection for integral action in the presence of single/multiple actuator saturation has been a subject of considerable research and is fairly mature.
Unfortunately, there is no analytical control design technique available for MIMO systems having output (or state) constraints. The majority of control system applications with output constraints in industrial applications are limited to SISO systems or MIMO systems with very little input-output interaction, i.e., all output constraints are dominantly affected by a single common actuator.